We develop the theory of canonical and pluricanonical adjoints, of global canonical and pluricanonical adjoints, and of adjoints and global adjoints to an irreducible, algebraic hypersurface V \subset P^n, under certain hypotheses on the singularities of V. We subsequently apply the results of the theory to construct a non-singular threefold of general type X, desingularization of a hypersurface V of degree six in P^4, having the birational invariants q_1=q_2=p_g=0, P_2=P_3=5. We demonstrate that the bicanonical map \varphi_{|2K_X|} is birational and finally, as a consequence of the Riemann-Roch theorem and vanishing theorems, we prove that any non-singular model Y, birationally equivalent to X, has the canonical divisors K_Y that do not (simultaneously) satisfy the two properties: (K_Y^3)>0 and K_Y numerically effective.
Adjoints and pluricanonical Adjoints to an Algebraic Hypersurface
STAGNARO, EZIO
2001
Abstract
We develop the theory of canonical and pluricanonical adjoints, of global canonical and pluricanonical adjoints, and of adjoints and global adjoints to an irreducible, algebraic hypersurface V \subset P^n, under certain hypotheses on the singularities of V. We subsequently apply the results of the theory to construct a non-singular threefold of general type X, desingularization of a hypersurface V of degree six in P^4, having the birational invariants q_1=q_2=p_g=0, P_2=P_3=5. We demonstrate that the bicanonical map \varphi_{|2K_X|} is birational and finally, as a consequence of the Riemann-Roch theorem and vanishing theorems, we prove that any non-singular model Y, birationally equivalent to X, has the canonical divisors K_Y that do not (simultaneously) satisfy the two properties: (K_Y^3)>0 and K_Y numerically effective.
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